January 12, 2018

Ur American Sh1thole Values NFL Open Thread

by Ugh

More Trump because why not? Eh. I've seen responses to the latest sharted eruption from POTUS saying that it "does not represent American values," to which my response of late is "Are you sure?" Almost 63 million people voted for this waste. There is a long history of American racism in immigration (and obviously other) matters that continues to this day, apparently. So, who are we kidding?

Meanwhile, the GOP is in all in on doing everything they can to deny African Americans the ability to vote. Full stop, not even pretending anymore. "Party of Lincoln" GFY.

Anyway, all is full game here. Rooting for a Pats-Vikings Superbowl here in suddenly balmy DC. I recommend Belize for a visit. The Southern Reach Trilogy was interesting and different, but not sure I'd recommend it.

Comments

if forced to choose would you rather live in Haiti or Norway?

I'm not cleek but I'll reply.

It would be a tough call. I have enough money that I could probably live pretty well in Haiti - far more so than in Norway, which is actually a pretty expensive place to live - and I hate hate hate the dark winters. Plus, I have a strong interest in Afro-Caribbean musical traditions, of which Haiti has one of the most interesting.

Was your question intended to make a point about immigration? Or was it just intended to be a defense of the POTUS' comments?

From what I've read he used a vulgar term to describe some countries that we all agree aren't great places to live, is that a fair summary?

That's my understanding. And basically it just doesn't bug me. I am personally no stranger to vulgar expressions, for good or ill, and for which I hope wj can forgive me.

The part that bugs me is the "Why are we taking people from sh*thole countries" that bugs me.

So, if we are interested in pursuing the issue of immigration, rather than the POTUS' vocabulary, perhaps that's what we should focus on.

There's also the fact that the countries under discussion have majority black populations, which reinforces the sense that the POTUS sees black people as inferior. It's of a piece with comments about people from Nigeria not wanting to go back to their "mud huts".

whether it's a good idea to take refugees from a country without discriminating based on skills and language and stuff like that.

Yes, it is a good idea. Paul Ryan recently complained that Americans aren't having enough babies. Infants don't speak English, and their resumes are thin. Also, people have to take care of them, rather than they taking care of other people. Maybe other solutions make more sense, like welcoming immigrants.

Well consider that, for example, a bigger percentage of immigrants from Nigeria (one of those sh*thole African countries, you know) have college degrees than the percentage of Americans who do. Which yes, does rather suggest that for Trump it is a skin color thing.

I don't believe we should prioritize poaching the best and brightest from other countries, unless they are fleeing obvious oppression. They should remain in situ and help build their economies so they can catch up with and surpass the United States, and China for that matter.

Who do we think we are?

Locking the less fortunate, who work their asses off, whether they are fleeing refugees or not, out of the United States only hurts us in the long run.

It's precisely the same as locking the steerage passengers on the Titanic below decks as the wealthy take the lifeboats.

If we do rape other countries of their best and brightest, merely to benefit ourselves, other countries should turn to China and Russia and other actors for assistance, including military.

China is making big inroads in Africa, shrinking our sphere of influence.

I don't believe we should prioritize poaching the best and brightest from other countries, unless they are fleeing obvious oppression. They should remain in situ and help build their economies so they can catch up with and surpass the United States, and China for that matter.

Who do we think we are?

If we were, as a nation, actively going out and recruiting them, you might have a point. But all we are doing is allowing them to choose to come here.

Yes, but now we are talking about NOT allowing the less fortunate and less educated in here, so the others have a leg up on the immigration list.

No, we don't actively recruit, but I'm pretty sure the American embassies, the Voice of America and suchlike extoll the virtues of living in exceptional America and the advantages of becoming U.S. citizens.

Maybe not anymore. Maybe when an educated, professional Ecuadoran starts nosing around with an eye to moving his or her family here, we say, "Nah, not a good idea. The U.S. is a shithole."

Yes, but now we are talking about NOT allowing the less fortunate and less educated in here, so the others have a leg up on the immigration list.

I'd offer to bet you that Trump can't imagine any of those immigrants from "sh*thole countries" being educated people who would get thru on a merit-based system. But I suspect I'm not winning any sucker bets tonight.

I'll take a wild guess that our latest playmate is a maternity-ward immigrant and not a port-of-entry immigrant. And proud of it, as if it's a personal accomplishment in some way.

He, Trump being a shining example of what maternity-ward immigrants can be like, I can't see it as evidence of superiority, myself.

"Shithole countries" is not the obscenity. "Take them out" is an obscenity. Deporting people who became Americans involuntarily (just like the maternity-ward kind) back "home" is an obscenity. Birtherism is an obscenity. Marketing Trump hotels to rich Russians who want to give birth in the US while railing about "anchor babies" is a hypocritical obscenity.

No, conservative death squads in el Salvador. And Costa Rica provides universal healthcare to its citizens and permanent residents, whereas, at the behest of American conservative influence, El Salvadorans may be in danger of losing their universal system, no doubt helped by the stray American-supplied bullets that find themselves embedded in liberal El Salvadoran skulls.

Tell you what, tell me what race you are and I might be able work up a bolus of prejudice against your people based on your representation of them here today.

Something tells me you are an outlying data point however, and generalizing from the particular in the conservative manner would be highly unfair.

It would be like believing ben carson is an asshole and then turning down taxi rides from all black drivers or brain surgery from all black surgeons based on his unfortunate choice to be an asshole, against his mother's best wishes for him.

Besides, there are few Irishmen is either El Salvador and Costa Rica, so what's to hate ethnically? Ha ha.

I don't have a transcript of Trump's comments. Assuming the reports are true - and at this point I have little reason to believe otherwise - they have as much to do with sensible immigration policy as "Excuse me, your water fountain is over there, Sir." has to do with proper hydration.

And I dunno how to quantify such things, but they both seem pretty obscene to me.

While I'm on about things I don't know...

What does incompetence and unfitness have to look like if this isn't it? 25.4 may as well be an impact-activated parachute.

And I dunno how to quantify such things, but they both seem pretty obscene to me.

Me too.

What's happening here is depressing beyond belief. Actual tears emerge at times. It has been this way since the evening of November 8, 2016. I'm coping by fixating on crazy little hobbies of mine. And checking in here, of course. And I worked on Virginia elections (yay!). And I will again in 2018 (!). Volunteering for local immigrants in addition to the usual 45 minutes a week of Meals on Wheels that I've done for years. Not much else that I have the creativity to do since I still have to work for a living.

Careful focus on stuff that is inherently interesting is how I'm getting through this. Everything matters.

I just damned the governess. Apologies for taking so long, but am in the hospital.

As for my take on shithole countries, it seems that anyone with any sense of history would realize that country becomes 'crappy' not because of the people who live in it, but because of larger historical factors. Given that Haiti (for example) is the first example of a successful slave revolution and was only permitted to remain free after they pay France 9 million gold francs.

Following Haiti's independence, former French slave-owners submitted detailed tabulations of their losses to the French government, with line items for each of "their" slaves that had been "lost" with Haitian independence. In 1825, the French King, Charles X, demanded that Haiti pay an "independence debt" to compensate former colonists for the slaves who had won their freedom in the Haitian Revolution. With warships stationed along the Haitian coast backing up the French demand, France insisted that Haiti pay its former coloniser 150m gold francs – ten times the fledgling black nation's total annual revenues.

Under threat of a French military invasion that aimed at the re-enslavement of the population, the Haitian government had little choice but to agree to pay. Haiti's government was also forced to finance the debt through loans from a single French bank, which capitalised on its monopoly by gauging Haiti with exorbitant interest rates and fees.

The original sum of the indemnity was subsequently reduced, but Haiti still disbursed 90m gold francs to France. This second price the French exacted for the independence Haitians had won in battle was, even in 1825, not lawful. When the original indemnity was imposed by the French king, the slave trade was technically illegal; such a transaction – exchanging cash for human lives valued as slave labour – represented a gross violation of both French and international laws. And Haiti was still paying off this "independence debt" in 1947 – 140 years after the abolition of the slave trade and 85 years after the emancipation proclamation.

One would think that the French would have wanted the Statue of Liberty to be built Port-au-Prince harbor, given that the Haitian people value it so much

A lawsuit launched by the Haitian government to recuperate these extorted funds was aborted prematurely in 2004, with the French-backed overthrow of the government that had had the temerity to point out that France "extorted this money from Haiti by force and… should give it back to us so that we can build primary schools, primary healthcare, water systems and roads".

Ah, yes, but that's the French you might say. But after Kennedy's assassination, the US propped up Duvalier because we hated Castro so much. So yes, Haiti may be a shithole country, but the reason for it is not because of the people, it's because of targeted foreign policies that have left the country broken and unable to cope. But if you believe that Trump was sensitive to those historical currents, go hang out at the America First! forum at reddit.

[snark]Give The Donald a chance to redeem himself by asking him, whether he would also welcome those tent-dwelling yodelling* reindeer herders with the funny hats from the Northern part of Norway, who are so superstitious that they never mention bears by name! They should be easy to retrain to herd Tru Merkin caribou, a ressource criminally underutilized due to Obama's sabotage of sound use(TM) of the Arctic**[/snark]

For the record, I have at times seriously considered moving to Norway for work (not just for the holidays as I used to).
At least they still have snow in the winter ;-)
All those digestive final product cavity lands mentioned would be far too warm for this heat sensitive German and replacing their population with Aryan inbreds* would even lower their appeal.

*btw, studied racists of the early 20th century considered Scandinavians to be racially spoiled since the Viking age for importing too many celtic and slavic slaves and not abstaining from producing offspring with them.

Thanks. Your advice is well received. I should find a way to channel the negative into something positive - before it devours me. But if I'm being honest, rage as a place-holder for hope feels necessary right now. I'm not sure I should (or want to) let it go.

russell, in the sense of latin 'nomen', i.e. term referring specifically to something/body. 'Bear' is a taboo word (I have even read that 'bear' itself is a term used by our indogermanic ancestors* to avoid the now forgotten 'true name' of said animal). To use it means calling the being itself. Ursine visitations are considered by most to be less than ideal events in most circumstances. Think Candyman in fur.

I assumed it was a joke but given that I was taking a short break from Latin homework when I read it and this (the contextual meaning of nomen) is a notorious stumbling block I could not keep myself from posting this pedantry. ;-)

CharlesWT, they would have a field day with the DNA survey of Iceland showing that the vast majority of women in the early days of settlement were celtic (=slaves taken from the British Isles by Norwegians on their way to Iceland). Same with St.Thomas Aquinas and the fact that all males were females in the beginning of embryonal development thus 'proving' that females are defective males that did not achieve completion as he claimed.
With a little bit of sophistry everything is possible (apology to the original sophists that got so unfairly maligned by Plato).

It's amazing the things that science can prove, when you already know what answer you want.

Today, we have eliminated the middleman. No appeals to science needed. (Indeed, the whole concept of science rejected.) So much quicker and easier to just assert your conclusion and sneer at anyone with the temerity to disagree.

This past Friday the SCOTUS accepted two more redistricting cases, these from Texas. I'm looking forward to all four decisions -- I love it when the generally innumerate Court has to deal with statistics.

The current Court majority has no problem abjuring historical and/or legal precedent or the clear meaning of English words, so political prejudice it is. No need to delve into philosophy, much less statistics.

The district court in Wisconsin has left them little choice. The court there didn't say that the efficiency gap was the right measure; they did say that some statistical test was a sufficient measure to determine unconstitutional gerrymandering. The efficiency gap applied to Maryland will say it's a gerrymander; applied to North Carolina, if that case is accepted down the road a bit, the efficiency gap will say it's a gerrymander.

If they won't accept statistics, then we're in for a steady stream of cases, because like pornography, "we'll know it when we see it."

The Cross-State Air Pollution Rule decision may be informative. There, the Court said that there is some cross-state formula that meets constitutional muster -- it might as well be this one, we can't keep sending it back to the DC Circuit court forever. Kennedy has indicated that there is probably a statistical standard that meets constitutional muster -- if not the efficiency gap, what?

At one point Terry does criticize Coates. It may be just Coates character and background, and should be accepted, but his identification with Baldwin rather than King, Malcolm, DuBois or Fanon is indicative, and needs criticism. Cornel West and Adolph Reed are absolutely right about him.

2) A little belatedly, I am revisiting the Russian Revolution. I will be reading Sheila Fitzgerald, but right now I cannot recommend more highly Alexander Rabinowitch slightly outdated trilogy on the Bolsheviks. It looks at 1917 in Petrograd day by day, even hour by hour, and exlains why and how the Bolsheviks won not only in terms of strategy and tactics, but especially in terms of contingencies and adaptation to forces outside of their control. And does so in a very readable even exciting prose.

"The July uprising was initiated in the First Machine Gun Regiment. Pinning down the precise time when plans for the rebellion began within the regiment itself is difficult, but it appears clear that this occurred well before the July 3 cabinet crisis often cited by Western and Soviet sources as one of the uprising's major precipitants. As was noted in the previous chapter, the Kerensky offen-
sive threatened many garrison units with immediate transfer to the front and only after the exertion of pressure from the Petrograd Soviet and the Bolshevik Party leadership was a soldiers' rebellion averted during the earliest days of the Russian advance. At that time members of the First Machine Gun Regiment canceled preparations for an immediate uprising, satisfying themselves with a repudiation of their Regimental Committee and a declaration of their refusal to fulfill further Provisional Government troop levies."

The soldiers in the Petrograd staging ground did not want to die in that fucking war for Britain, America, and France. Kerensky started an offensive June 19, but Kerensky, if he wanted a democratic Russia that was not a pariah to the economic West, really had no choice. But this is what lost the revolution.

Lenin and the Bolsheviks were cautious and reluctant, but also had no choice but to follow the soldiers (who by this time, with millions dead, were mostly rural peasants, so radicl land reform, opposed by Kerensky and SR, was also critical.)

Anyway, the applications of 1917 are entirely about adaptability and using local and contingent conditions. That is the point.

(and the toleration for Forever War is another reason I am no longer a Democrat)

Of course the liberals, bourgeoisie and Social Democrats cheered Kerensky like crazy on June 19th.

Those who dislike my attitude toward violent revolution should deal with a simple question: Which side would you have been on in Petrograd in 1917? Would you have sided with Kerensky yo ship another million to Galicia to die for foreign relations and Imperial power-sharing? Or would you have been with the armed soldiers and sailors, knowing that resistance would mean violence and civil war?

From what I read, most here would have shot those deserting country boys dead in the street?

This is not ancient history. Question came up round 1968 and 1972, and the question is alive now in demands to support the war mongering Democrat or hand the country to Republicans.

Yeah, I read LGM. This was fantastic, as good an article on Martin Luther King as I have ever read.

Brandon Terry on MLK

I just finished it. Highly recommended.

As to 1917 Petrograd, I have no clue what side I would have taken, and in some very important ways, it is not the correct question. That said, Kerensky pretty much ended any chance of a democratic Russia by trying to force a broken country to continue the tragic, utterly futile, and destructive war with Germany.

Armed revolution may have been a political mistake, but the essential justice of taking up armed resistance at that time and under those circumstances has a good deal of merit.

As for history of that era, I am still favorably disposed toward the works of that old Trot, Issaac Deutcher.

Meanwhile, 1/4 remains the correct answer for 3 random points ON the circle

Let P,Q and R be arbitrary points in the disk and P',Q' and R' be their respective nearest orthogonal projections on to the circle. Then triangle PQR contains the disk's center C if and only if triangle P'Q'R' does.

("Only if" is almost trivial. "If" requires showing that if PQR does not contain C there exists a bisection of the disk with P,Q and R all in the same half, implying P',Q' and R' are in that half as well.)

Hence the probability that PQR contains C is the same as the probability that the triangle formed by three arbitrary points on the circle contains C, namely, 1/4.

At January 14, 2018 at 03:41 AM I joked that there would be an attempt to redeem Trump by referring to the Sami minority in Norway.
Looks like the new head of the DHS did just that by denying that Norway was an all-white country. Nothing about funny hats and reindeers though, alas!

Hence the probability that PQR contains C is the same as the probability that the triangle formed by three arbitrary points on the circle contains C, namely, 1/4.

This seems to me to imply that a given point in the same plane as any three arbitrary points in that plane would have a 1/4 probability of being in the triangle formed by those three arbitrary points.

Take the given point as the center of a circle in that plane with a radius equal to or greater than the distance to the furthest of the three arbitrary points and you've got yourself the same circle question with the same answer. You just have to make the circle as big as it needs to be.

(Unless I'm missing something. And I do have a non-zero probability of missing something.)

Intuitively, the probability of hsh's 3:24 points on a plane doesn't have a shot at being 1/4, while points on the circumference triangle does seem to. But my brain abandoned this particular problem way up thread.

Look at it this way - a plane is infinite, so you can arbitrarily consider any point on it as the center. Take three points at random. Pick two from which to form two lines going through the third. Those two lines will form an angle between 0 and 180 degrees. (Well, two angles, on either side of the third point, but we're concerned with the one on the side opposite the first two.)

The average angle over the even probability distribution from 0 degrees to 180 degrees is 90 degrees. A 90-degree takes up 1/4 of the plane.

To finish that thought, a fourth point would have to be anywhere within the angle formed by projecting lines from the first two points through the third point in order for the third point to be within the triangle formed by the other three points (i.e. first, second, and fourth).

Of course, mine isn't really any different from Tony P.'s, except he was looking at one point being the center of a circle and the other three on the circumference. But it's the same thing, whether they're on the circle, inside the circle, or just in the same plane.

The notion of choosing a point at random in the plane doesn't make sense. You can put a uniform probability density function on a disk; you can 't do so on the plane.

If your p.d.f. is radially symmetric about a given point C, then the probability of your triangle containing C will still be 1/4. (In this case it does make sense to think of C as the "center" of the plane.) Lacking this symmetry, it seems very likely that the probability is something different.

But, but, the triangle in the plane only takes up 1/4 If you assume the first two points are at the infinite edge of the plane. It us true the fourth points probability of being in the angle would be 1/4, but not in the triangle. The citcle is bounded but the plane is not.

I think Ufficio may be right about the uniform probability. (The non-zero probability that I might be missing something applies here.)

But, that aside, the points don’t have to be at the “edge” of the plane (or, better yet, the circle). The lines extend to infinity (or at least to the edge of the circle if that’s all we care about).

The fourth point isn’t the one that has to be inside the triange. It’s one of the vertices of the triangle, along with the first two points. The third point (or center of the circle) is the one (possibly) inside the triange. It’s just that fourth point has to be inside the angle (or sector of the circle (as opposed to triangle)) formed by the two lines going through the third point (or center) and the first two points for the third point (or center) to be inside the triangle formed by the first two points and the fourth.

A bill to avert a Federal government shutdown is needed by Friday. Democrats want something done about DACA. Among other things.

So far, what the Republican leadership appears willing to give, in exchange for Democratic votes, is a CHIP renewal -- which is also among the things that Democrats want done. For what is, quite simply, just a one month spending authorization.

So answer me this. Why not take the CHIP renewal? And catch DACA in the next couple of weeks? Yeah, it would be good to get both now. But sometimes it makes sense to take half a loaf now, and the other half tomorrow.

Because the Dems are sick of getting pushed around and are willing to say screw you/have been emboldened by [insert developments here] and now want to take advantage of what they view as an ideal time to get back at the Republicans?

How one spins that probably depends on which side you are on, and I'd aim for the former rather than the latter.

This pitched battle is over a mere $15billion/year. In other words, peanuts. The GOP, claiming they support the program before letting it die, could authorize it without one Democratic vote.

Since the GOP can't get it's act together and pass stuff (like funding the 'effing government) on their own, the Dems have some leverage not generally accorded the minority party. Therefore they feel they should push their position to the max, and now is the time to do so, not later....because as you know, things change.

Some wag once pointed out that infinity gets really big toward the end.

When you pick 3 random points "on the plane", the very fact that you "picked" them implies that they are separated by finite distances. That means they define a triangle of finite area. "The plane" has infinite area. Does that have some bearing on the probability=1/4 question?

Two things you might enjoy:

1) This 3blue1brown video which gives an off-beat solution of the 3-points-on-a-circle problem -- as a mere preliminary to the analogous 4-points-on-a-sphere problem.

2) This proof that "every number is interesting":
Assume a least-interesting number exists.
That makes it interesting.
QED

It's as well to steer clear of arguments involving infinity if you can. In particular, how do you choose a random point from an infinite range?

The straightforward argument for a triangle inside the circle is to move the vertexes to the nearest point on the perimeter (ie move them outwards along a radius). This operation moves each side of the triangle away from the centre. Since the sides cannot cross the centre as they move, the centre remains inside or outside the triangle.

Now refer to the solution for three points on the perimeter of a circle.

(I think someone said something like this already.)

Of course, one can draw any size of circle, to contain any triangle, so in the end the circle exists only to provide a definition of a random point.

The benefit of my misguided musings is that they led me to clearly visualize the problem in a way I hadn't before. What matters is not how far the points are from the center of the circle, but their angular positions.

Do you have to take a mathematical discussion down to the lowest common denominator?

I should give Tony P. and Uffico credit for getting me 90% of the way to a full understanding of the problem. I wouldn't have gotten there on my own, at least not nearly as quickly (defining "quickly" very loosely).

I am still puzzled, probably because of what pro bono says about infinity. Jaynes says in his probability book that I haven’t read that you should always reason from finite cases and then carefully define how you go to infinity if you do that at all.

Anyway, if you pick three random points in a plane A, B, and D and ask for the chance that a fourth point C ( for center when we later invoke circles) is inside it, the answer is zero stated that way because any triangle is finite and the plane is infinite. But if you use the point from set A,B, and D which is furthest from C to define the radius of a circle then maybe the argument for 1/ 4 goes through, if you limit yourself to points in the circle and say that the probability of getting a point on the circumference is equal to the probability of a point on the radius that goes from C out to the circumference. Not happy with that.

I suppose it is a symmetry argument. You assume that he pdf inside the circle is uniform over the area and, well, I am not completely sure in my head that a uniform distribution over the circumference lets you slide points along the radius and make the same argument but I guess because of symmetry it might. Have to talk myself into it.

I mean, you are going from a probability per length to a probability per area if you invoke the interior. If you define points on a given radius as being equivalent in its probability to the point on the circumference then it is in effect still a density defined in terms of length ( or angle).

But maybe the symmetry gets around that. Yep, still confused. I have to do some things though, so maybe I will come back later.

As far as the infinite plane is concerned, I visualized it as first picking two of the vertices of the triange. Then picking the point potentially to be inside the triangle yet to be defined. Once you have that, you have an angle of x degrees projecting outward from the third point. That angle is infinite in area, so the way I was looking at it, a fourth point picked to finish the triangle has a probability of x/360 of ending up inside that angle.

This may well be nonsensical when considering an infinite plane, but I think it’s utterly sensible when considering a finite circle. It’s also why the radial positions of the triangle’s vertices don’t matter, only their angular positions. You end up considering the same angle regardless of where the first two vertices are radially. And that angle projects out from the center to the outer circumference, opposite the first two vertices, so the third vertex can be anywhere in that sector of the circle for the center to be inside the triange.

If you like, forget the circle and just draw three radial lines from the centre point, at random angles. Then choose three points at arbitrary distances along the three radial lines (if you like you can use a random distribution weighted so that the points are equally distributed by area, up to some maximum radius). Draw a triangle between these three points.

Look at the three angles between the three radial lines going clockwise (or widdershins, if you prefer). If all three angles are less than 180 degrees, the triangle includes the centre. If one is more than 180 degrees, it doesn't.

What's the probability that none of the three angles is greater than 180 degrees? Look at the (smaller) angle between the first two radial lines you drew. If the angle between them is vanishingly small, the probability is zero. If the angle is arbitrarily close to 180 degrees, the probability is half. For angles in between, it varies linearly between zero and half, so on average it's a quarter.

I thought of anither way to formulate what is bugging me. Take an area. Or rather, take several different areas, one of which is a circle. For each of these areas, pick three random points inside the area which will define a triangle. Now do this a zillion times Monte Carlo style. What is the average area of the triangle expressed as a fraction of the total area? That is a proxy for asking what is the chance that a triangle formed by three random points will contain a fourth random point.

The areas have to be convex, if I am using the term correctly. I mean the three random points must form a triangle whose area is contained inside the larger area. Would you get the same answer for various shapes? Would it be 1/4? If so, then you say that the chance that three arbitrary points define a triangle which contains a fourth is 1/4.

If I understand things correctly, which is not at all certain, the arguments in this thread depend on the fourth point being treated as the center of a circle. But if all four points are picked at random in a predefined area does the 1/ 4 answer still hold?

Thought about it further. Pick out one point. Then pick three at random to define the triangle. Is the chance that the first point can be found in the triangle going to change as you move towards the perimeter of the area? Yeah. Seems intuitive. If it was very close to the perimeter it drops towards zero. The center should have the best chance, and so the average area of the triangles should be less than 1/ 4.

If that is right then if you want to talk about entire planes I suppose t makes sense to imagine a given point as the center of a gigantic circle.

The original problem was about the probability of the center of a circle being inside a triangle formed from 3 points randomly chosen on (or in) that circle. It wasn't about a fourth random point being inside a random triangle.

It was my offshoot of that problem that concerned 4 random points on a plane (since there's no true center).

Just a minor point: three random points on a plane define a circle (where all three points are on the circumference).

Because there's no preferred coordinate system, you can always move the circle so that the center is at the origin (x=0,y=0), rotate so that "point #1" is on the x-axis, and scale (or define length units), so that the circle is of radius=1.

I know, but the extension to the plane is what got me obsessing about it. The problem is defining what it would mean to pick random points in an infinite plane. You have to define an appropriate finite size problem, I think, and then extend it if I understand Jaynes correctly.

I think one needs a space where every point is the same. You could have a unit square where the point ( x,y) = (x +n, y+m) where n and m are integers so it is a flat space which wraps around on itself, but I am not smart enough to do the problem. Or one could think of picking three random points in a surface of a sphere, forming a spherical triangle and asking what the average area would be, but then it is non Euclidean. And I know nothing about spherical triangles.